Turan Type Inequalities Research Articles

Lq-norm inequalities for the generalized derivative of a polynomial

In this paper, we aim to present some Turan-type inequalities in Lq-norm for the generalized derivative of a polynomial. We further extend these results to the generalized polar derivative. Our results besides generalizing various known results also include other inequalities as special cases.

Some Turan-type inequalities for a class of rational functions with prescribed poles

In this paper we prove some results for rational functions with t-fold zeros at the origin and thereby obtain generalizations and refinements of some known inequalities for rational functions. These results in particular yield the improvement and generalizations of some polynomial inequalities.

Generalized Turan-type Inequalities for Polar Derivative of a Polynomial

Let $P(z)=a_0+\sum\limits_{\nu=\mu}^na_{\nu}z^{\nu}$, $1\leq\mu\leq n$, be a polynomial of degree $n$ having all its zeros in $|z|\leq k$, $k\geq 1$. We obtain an improvement and a generalization of an inequality in polar derivative proved by Somsuwan and Nakprasit [1]. Further, we also extend a result proved by Chanam and Dewan [2] to its polar version. Besides, our results are also found to generalize and improve some known inequalities.

Turan type inequalities for rational functions with prescribed poles

In this paper, we establish some inequalities for rational functions with prescribed poles having t-fold zeros at the origin. The estimates obtained generalise as well as refine some known results for rational functions and in turn, produce extensions of some polynomial inequalities earlier proved by Turan, Jain etc.

Comparison inequalities between rational functions with prescribed poles

Various versions of Bernstein and Turan-type inequalities are a classical topic in analysis. Over a period, these inequalities have been generalized for different classes of functions. Here, we establish some new comparison inequalities for rational functions in the complex plane that are inspired by some classical inequalities of Bernstein and Turan type. The obtained results produce extensions of many inequalities for polynomials as special cases.

On Generalized Complete (p, q)-Elliptic Integrals

In this paper, we study the generalized complete (p, q)-elliptic integrals of the first and second kind as an application of generalized trigonometric functions with two parameters, and establish the monotonicity, generalized convexity and concavity of these functions. In particular, some Turan type inequalities are given. Finally, we also show some new series representations of these functions by applying Alzer and Richard’s methods.

Functional Inequalities and Bounds for the Generalized Marcum Function of the Second Kind

In this paper, we consider the generalized Marcum function of the second kind as an analogous function of the so-called generalized Marcum Q-function. We provide the log-convexity (log-concavity) property for its unit complement and improve some of our previous results on it. One of the transformed functions of the generalized Marcum function of the second kind is discussed in details in this paper. This form of the generalized Marcum function of the second kind supplies various important inequalities. We also discuss the Turan type inequality for the generalized Marcum Q-function. Additionally, we provide the bounds for the generalized Marcum function of the second kind as well as for its symmetric difference.

Functional estimates and integral inequalities for the Fox–Wright function

In this paper, our aim is to show some mean value inequalities for the Fox–Wright function. In order to prove our main results, we present some monotonicity, convexity and concavity properties for some classes of functions related to the Fox–Wright function, which are in fact equivalents to the corresponding Turan type inequalities for this function. As a direct consequences, it deduces some new results involving some special functions, such as the generalized hypergeometric function, the four parameters Wright function and three parameter Mittag–Leffler type function. At the end of this paper, several integrals inequalities associated to the Fox–Wright function are established. As applications, new functional inequalities (such as Turan-type inequalities) for the incomplete Fox–Wright and incomplete generalized hypergeometric functions are established.

On Turan-type inequalities for trigonometric polynomials of half-integer order

Some exact inequalities of the Turan type are obtained in the paper for trigonometric polynomials $h(x)$ of half-interger order $n+\frac {1}{2}$, $n=1, 2, ...$, such that all $2n+1$ their zeros are real and located on a segment $[0;2\pi )$. Namely, the inequality that relates the norms in the space $C$ of the trigonometric polynomials $h(x)$ of half-integer order $n+\frac {1}{2}$, $n=1, 2, ...$, and its second derivative $h''(x)$, $\|h''\|_c\ge \frac {2n+1}{4}\|h\|_c$, that is the inequalities that connect the norms in the space $L_2$ of the trigonometric polynomials $h(x)$ of half-interger order $n+\frac {1}{2}$, $n=1, 2, ...$, and its first derivative $h'(x)$, that is $\|h'\|_{L_2}\ge \sqrt {\frac {2n+1}{8}}\|h\|_{L_2}$. The resulting inequalities cannot be improved. In proving the theorems, we use the method that was developed by V.F. Babenko and S.A. Pichugov for trigonometric polynomials, all of whose roots are real.

Generalizations of Bernstein and Turán-Type Inequalities for the Polar Derivative of a Complex Polynomial

The goal of this paper is to extend to the polar derivatives some previous inequalities between the $$L^\gamma $$-norms of the derivative and of the polynomial itself, in case when the zeros are outside or inside some closed disk. Apart from this, our results also derive polar derivative analogues of some classical Bernstein and Turan-type inequalities that relate the sup-norms of a polynomial to that of its derivative on the unit circle.

Monotonicity patterns and functional inequalities for classical and generalized Wright functions

In this paper our aim is to present the completely monotonicity and convexity properties for the Wright function. As consequences of these results, we present some functional inequalities. Moreover, we derive the monotonicity and log-convexity results for the generalized Wright functions. As applications, we present several new inequalities (like Turan type inequalities) and we prove some geometric properties for four--parametric Mittag--Leffler functions.

Functional inequalities for the Fox–Wright functions

In this paper, our aim is to establish some mean value inequalities for the Fox–Wright functions, such as Turan-type inequalities, Lazarevic and Wilker-type inequalities. As applications we derive some new type inequalities for hypergeometric functions and the four-parametric Mittag–Leffler functions. Furthermore, we prove the monotonicity of ratios for sections of series of Fox–Wright functions. The results are also closely connected with Turan-type inequalities. Moreover, some other type inequalities are also presented. At the end of the paper, some problems are stated which may be of interest for further research.

Turán Type Inequalities for Jacobi-Dunkl Polynomials

In this paper we establish Turan type inequalities for Jacobi-Dunkl polynomials. These complete an earlier result proven by G. Gasper in [1], for Jacobi polynomials.

Logconcavity, twice-logconcavity and Turán-type inequalities

This paper presents a brief review of the logconcavity of some compound distributions and its relationship to Turan-type inequalities. Further, we introduce a new concept of twice-logconcavity (twice-logconvexity) and show that many discrete probability distributions and combinatorial sequences are twice-logconcave. These results are closely related to other areas of mathematics such as combinatorics and analysis.

Functional Inequalities for the Mittag–Leffler Functions

In this paper, some Turan-type inequalities for Mittag–Leffler functions are considered. The method is based on proving monotonicity for special ratio of sections for series of Mittag–Leffler functions. Furthermore, we deduce the Lazarevic- and Wilker-type inequalities for Mittag–Leffler functions.

(p, q)-Extended Bessel and Modified Bessel Functions of the First Kind

Inspired by certain recent extensions of the Euler’s beta, Gaus hypergeometric and confluent hypergeometric functions (Choi et al. in Honam Math 36(2):339–367, 2014), we introduce (p, q)-extended Bessel function \(J_{\nu ,p,q}\), the (p, q)-extended modified Bessel function \(I_{\nu ,p,q}\) of the first kind of order \(\nu \) by making use two additional parameters in the integrand, as well as the (p, q)-extended Struve \(\mathbf{H}_{\nu ,p,q}\) and the modified Struve \(\mathbf{L}_{\nu ,p,q}\) functions. Systematic investigation of its properties, among others integral representations, bounding inequalites Mellin transforms (for all newly defined Bessel and Struve functions), complete monotonicity, Turan type inequality, associated non-homogeneous differential-difference equations (exclusively for extended Bessel functions) are presented. Brief presentation of another members of Bessel functions family: spherical, ultraspherical, Delerue hyper-Bessel and their modified counterparts and the Wright generalized Bessel function with links to their (p, q)-extensions are proposed.

On some inequalities involving Turán-type inequalities

AbstractUsing a new form of the Cauchy–Bunyakovsky–Schwarz inequality, we prove inequalities involving Turan-type inequalities for some special functions.

Turán type inequalities for general Bessel functions

In this paper some Turan type inequalities for the general Bessel function, monotonicity and bounds for its logarithmic derivative are derived. Moreover we find the series representation and the relative extrema of the Turanian of general Bessel functions. The key tools in the proofs are the recurrence relations together with some asymptotic relations for Bessel functions.

Treatise on Generalized Krätzel Function

The paper deals with complete monotonicity and log-convexity of the type-1 and type-2 generalized Kratzel functions. The Laguerre type and Turan type inequalities for these functions are also substantiated. The generalized Kratzel functions of both types were brought in closed form via Fox’s H-function and existence conditions are given. The asymptotic behavior of these functions at zero and infinity were deduced and methods for the evaluation of some integrals involving the generalized Kratzel function are indicated.

Weighted Turán type inequality for rational functions with prescribed poles

(Communicated by I. Rasa) Abstract. Firstly, we introduce a new type of weight functions named as N-doubling weights, which is an essential generalization of the well known doubling weights. Secondly, we establish a weighted Turan type inequality with N-doubling weights and a Nikolskii-Turan type inequality for rational functions with prescribed poles. Our results generalize some known Turan type inequality both for polynomials and rational functions.